Riemann curvature tensor on [formula omitted] spaces and possible applications

Abstract

We show that, on every RCD space, it is possible to introduce, by a distributional-like approach, a Riemann curvature tensor.Since, after the works of Petrunin and Zhang–Zhu, we know that finite dimensional Alexandrov spaces are RCD spaces, our construction applies in particular to the Alexandrov setting. We conjecture that an RCD space is Alexandrov if and only if the sectional curvature – defined in terms of such abstract Riemann tensor – is bounded from below.